Diffusion Reaction Interaction for a Pair of Spheres - ETD ...

1953). The reaction rate for this case Ω 1 ( R1 for 1 0 → λ and 2 0 → λ ) in spherical
coordinates (r1,Θ 1, φ) has the form
Ω
1
= 2πDc
a
0
π
2
1 ∫
0
⎛ ⎞
⎜∂κ
⎟
⎝ ∂r1
⎠
r = a
1
1
sin Θ
1
dΘ
1 , 1 a1
r = , (3.41)
where D is the diffusion coefficient, c0 concentration infinitely far from the sphere, a1 is
the diameter of sink 1, κ refers to the dimensionless concentration defined by equation
(3.5), and spherical coordinate Θ1, from Figure 2.1.
To express the reaction rate we must first determine the form of the dimensionless
concentration in bispherical coordinates (Morse and Feshbach, 1953). Solving the
Laplace equation (3.1) the dimensionless concentration is
1
1
μ j
μ j
2
2
κ cosh μ cosη∑
Aj
e B je
Pj
( cosη)
j 0
∞ ⎛ ⎞
⎛ ⎞
⎛ ⎜ + ⎟ − ⎜ + ⎟ ⎞
⎝ ⎠
⎝ ⎠
= − ⎜ + ⎟ , (3.42)
⎜
⎟
= ⎝
⎠
Pj is the Legendre polynomial of the first kind of order j , and Aj and BBj are constants to
be determined by the boundary conditions. After applying the boundary conditions
(3.39) and (3.40) for infinitely fast reactions ( λ → 0)
to equation (3.42), the
dimensionless concentration in bispherical coordinates becomes,
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